Heisenberg uniqueness pairs and the Klein-Gordon equation

TL;DR

This paper introduces the concept of Heisenberg Uniqueness Pairs (HUPs) and characterizes when collections of exponentials are fundamental on a hyperbola, solving related problems in harmonic analysis and algebra density.

Contribution

It defines HUPs and provides a complete characterization for hyperbolas with equally spaced frequencies, linking to algebraic density problems.

Findings

Complete characterization of HUPs for hyperbolas with lattice-cross frequencies

Solution to a problem on algebra density generated by inner functions

Establishment of conditions for weak-star fundamentality of exponential collections

Abstract

The notion of a Heisenberg Uniqueness Pair (HUP) is introduced. This amounts to asking which collections of exponentials are weak-star fundamental in $L^\infty$ on a planar curve. In the case when the curve is a hyperbola, we can give a complete answer if the frequencies are restricted to equally spaced points on a lattice-cross. As a consequence, we solve a problem on the density of algebras generated by two inner functions raised by Matheson and Stessin.